Evidence That Hyperconjugative Interactions Are Not Responsible Or

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Computational evidence that hyperconjugative interactions are not responsible for the anomeric effect Yirong Mo*

The ‘anomeric effect’ is the thermodynamic preference for polar substituents to occupy the axial position in the chair conformation of various heterocycles. The most common explanation given for this effect at present is hyperconjugation from the lone pairs on the ring heteroatom to the antibonding orbital between the anomeric carbon and its linking substituent. Alternatively, the anomeric effect could be explained by intramolecular electrostatic interactions between local dipoles. Few models can provide convincing data for either theory at the quantum-mechanical level. Now, using the extended block-localized wavefunction method, which is the simplest form of valence bond theory, we have evaluated the degree of hyperconjugation in various compounds that display the anomeric effect and have interpreted their conformational preferences in terms of steric, hyperconjugation and dispersion effects. The results provide strong evidence that hyperconjugative interactions are not responsible for the anomeric effect and that it is better interpreted in terms of electrostatic interactions.

he ‘anomeric effect’ is a stereoelectronic effect that was discov- bond14. However, this has been refuted by Perrin and colleagues, ered by Edward1 in 1955 while studying carbohydrates; the who showed that the electrostatic model can rationalize the geo- Tterm was later coined by Lemieux and Chu¨2 in 1958. It metrical variations at least in 2-methoxy-1,3-dimethylhexahydro- refers to the thermodynamic preference for electronegative (polar) pyrimidine15. Further reﬁnements of this electrostatic model ... 16 substituents bonded to C1 (the anomeric carbon shown in Fig. 1 include the consideration of favourable C2H O hydrogen bonds . in a substituted tetrahydropyran) to take up an axial position (a- An alternative and popular rationalization of the anomeric effect anomer) rather than an equatorial position (b-anomer) in the is based on electron delocalization from the oxygen lone pairs to the 3 17 chair conformation of a monosaccharide . The anomeric centre is vacant antibonding orbital sCY* (Fig. 2b) . This kind of n s* critical to the reactivity of carbohydrates because it is the site at negative hyperconjugative interaction can explain the geometry which ring opening occurs, producing the important functional changes, because the electron delocalization increases the double- carbonyl group. The preference for the a-anomeric diastereomer bond character of the O2C1 bond but weakens the C12Y bond. increases as the electron-withdrawing capability of the anomeric sub- The hyperconjugation model, however, is inconsistent with the stitutent Y increases. When Y is a carbon atom substituent such as a reverse anomeric effect, where the equatorial conformation is pre- methyl group, however, the b-anomer is favoured so that steric ferred5,18, and some researchers have questioned the validity of the repulsions between the ring and the substituent are minimized. n s* hyperconjugation model16,19. A compromise that takes into Although the anomeric effect was initially found in carbo- account such different views is that both the dipole–dipole and hydrates and ubiquitously exists in monosaccharides and their hyperconjugative interactions contribute to the anomeric effect18. derivatives, the concept has now been generalized to saturated Although extensive computational studies have correctly heterocycles and acyclic systems containing heteroatoms. Many described and predicted the conformational features of compounds experimental and theoretical studies have been conducted regarding exhibiting the anomeric effect, few have been able to differentiate the understanding and utilization of this peculiar effect4–9, which various factors such as steric (Pauli exchange), electrostatic and elec- plays a central role in conformational analysis in modern organic tronic interactions at the quantum-mechanical level. Individual esti- chemistry10. mates of various interactions can provide deep insights into the The most intriguing issue surrounding the anomeric effect is why origin of the anomeric effect. Perrin and colleagues examined the it occurs. When Edward1 proposed the anomeric effect, he intui- change in the anomeric effect when the oxygen of an anomeric mol- tively linked it to the lone pairs on the ring oxygen. This idea ecule was replaced by nitrogen15. Owing to its lower electronegativ- later developed into the electrostatic model, which states that the ity when compared with oxygen, nitrogen would have weaker preference for the a-anomer arises from the favourable local electrostatic interactions and stronger hyperconjugative inter- dipole–dipole interaction, as illustrated in Fig. 2a (ref. 11). This elec- actions. The results revealed that the anomeric effect is weaker in trostatic model is in agreement with the experimental evidence, the nitrogen analogue and that 2-aminotetrahydropyran exhibits which suggests that aqueous solvation effects stabilize the b- the reverse anomeric effect20, suggesting that electrostatic inter- anomer and thus reduce anomeric stabilization in many systems actions predominate. However, the hyperconjugation model is greatly such as tetrahydropyranosyls12,13. It has also been claimed that the supported by direct computations of the n s*ands s* electrostatic model is unable to explain the variations in bond dis- stabilization energies based on the natural bond orbital (NBO) tances and angles around the anomeric carbon atom, such as the method21. It has been pointed out, however, that the NBO shortening of the O2C1 bond and the lengthening of the C12Y method tends to highly overestimate stabilizing electron

Department of Chemistry, Western Michigan University, Kalamazoo, Michigan 49008, USA. *e-mail: [email protected]

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O O methods are computationally demanding due to the non-orthogon- Y C1 C1 ality of orbitals, the introduction of doubly occupied (a feature of molecular orbital (MO) theory) but strictly localized (a feature of Y VB theory) bond orbitals can significantly simplify the computational process. Following this strategy, the wavefunction of an elec- Figure 1 | Anomeric effect in substituted tetrahydropyran. The anomeric tron-localized reference Lewis structure can be derived by effect refers to the preference for polar substituents to occupy the axial, partitioning the system into several mutually interacting subgroups rather than equatorial, position in a chair conformation. This is illustrated and restricting the expansion of each MO in only one subspace. All here with the thermodynamic equilibrium between a (right side) and b (left block-localized MOs are self-consistently optimized30–35. Our block- side) anomers of a substituted tetrahydropyran where Y is an localized wavefunction (BLW) method can not only decompose the electronegative substituent. intermolecular interaction energy into a few physically meaningful terms36, but can also probe the electron delocalization within a mol- O O ecule. Similar approaches have been successfully applied by others to 23,37–41 C C study conjugation and hyperconjugation effects . 1 1 In this work, we studied molecules that exemplify the anomeric Y Y effect, including dimethoxymethane and substituted tetrahydropyrans, as well as the reference molecules dimethyl ether, tetrahydro- ab pyrans and substituted cyclohexanes, which have no anomeric effect. The BLW method was applied to these systems to quantify Figure 2 | Two most common explanations for the anomeric effect. a,The the electron delocalization and steric effects and subsequently criti- electrostatic model states that the preference for the -anomer arises from a cally examine the role of hyperconjugative interactions in the favourable local dipole dipole interactions. b, The hyperconjugation model 2 anomeric effect. is based on the stabilization gained from electron delocalization from the oxygen lone pairs to the vacant antibonding orbital sCY*. Results delocalization energies (DEs) due to the non-optimization of bond Dimethoxymethane. Dimethoxymethane is the simplest example orbitals, as exemplified by the recent controversy over the nature of with which to illustrate the anomeric effect: its gauche the ethane rotation barrier22–26. conformation around the C2O bonds is more stable than the One plausible way by which to determine the impact of conjuga- trans conformation. Moreover, as there are two anomeric centres, tive and hyperconjugative electron delocalization is to use ab initio the most stable conformer, namely the gauche2gauche (GG) valence bond (VB) theory27,28, which can define wavefunctions for conformer shown in Fig. 3 (1a), is 3–5 kcal mol21 more stable individual resonance structures. Among all possible resonance than 1c the trans2trans (TT) conformer; this effect is about twice structures with electrons strictly localized on bonds or atoms, the as big as that in sugars. Dimethoxymethane has therefore been one with the lowest energy is taken as the reference to measure widely selected as the prototype for computational studies on the the electron delocalization effect29. Although ab initio VB anomeric effect19,42–44.

1a − GG 1b − GT 1c − TT

2a − α 2b − β

3a − α 3b − β

Figure 3 | Electron density difference (EDD) maps showing electron delocalization within compounds displaying the anomeric effect. a–c, Electron delocalization between the oxygen atoms and methyl groups in dimethoxymethane (GG, GT and TT conformers, isodensity 0.003 a.u.) (a), the oxygen atom and substituted groups in 2-ﬂuorotetrahydropyran (a-andb-anomers, isodensity 0.002 a.u.) (b) and the oxygen atom and substituted groups in 2-methyltetrahydropyran (a-andb-anomers, isodensity 0.002 a.u.) (c). The red colour denotes a gain and the blue colour a loss of electron density.

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Table 1 | Absolute energies and relative and delocalization energies for various conformers of dimethoxymethane.

Del Del 21 Basis Set Conformer E(C ,MP2) E(C ,HF) E(FL , BLW) DE* (kcal mol ) 6-31 þ G(d) GG 2268.71788 2267.95848 2267.87335 53.42 GT 3.37 2.32 2.82 53.92 TT 7.52 5.52 6.48 54.38 6-311 þ G(d,p) GG 2268.88891 2268.03024 2267.93984 56.72 GT 2.74 2.07 2.94 57.59 TT 6.56 5.20 7.02 58.55

21 Del For GG, E is given as absolute values in Hartree; for GT and TT, E is given as values relative to GG in kcal mol . *DE ¼ E(FL,BLW)2E(C ,HF).

Table 2 | Absolute energies (a.u.) of a-anomers and relative energies of b-anomers and the delocalization energies in substituted tetrahydropyrans (C5OH9Y, Y 5 F, OH, Cl, NH2 and CH3).

Del Del 21 YAnomerE(C ,MP2) E(C ,HF) E(FL , BLW) DE (kcal mol ) F a 2369.90446 2368.89261 2368.83684 35.00 b 3.41 2.80 3.46 35.66 OH a 2345.91027 2344.88796 2344.83107 35.69 b 1.32 0.77 2.19 37.11 Cl a 2729.90042 2728.92888 2728.87242 35.43 b 2.69 2.46 1.76 34.73

NH2 a 2326.06039 2325.05242 2324.99628 35.23 b 22.73 23.07 22.45 35.84

CH3 a 2310.02913 2309.05549 2309.00804 29.77 b 23.30 23.48 22.31 30.95

We investigated three stationary structures of dimethoxy- with the 6-311 þ G(d,p) basis set the DE values for conformers GG, methane—GG, GT (gauche–trans) and TT—the geometries of GT and TT are 56.7, 57.6 and 58.6 kcal mol21, respectively, indicat- which were optimized at the MP2/6-31 þ G(d) level (second- ing that each trans structure is nearly 1 kcal mol21 more stable than order Møller–Plesset (MP2) theory is an electron-correlated the gauche structure as a result of electron delocalization. This is method compared with the Hartree–Fock (HF) approximation). completely different from the popular hyperconjugation model, Of these structures, GG represents the ground state, but all show suggesting that the overall stability of the GG conformer relative no imaginary frequency. Table 1 presents a compilation of the rela- to the TT conformer must not result from hyperconjugative inter- tive MP2 and HF energies of these three conformers with the basis actions. It should be stressed that when we claim that hyperconjugation sets of 6-31 þ G(d) and 6-311 þ G(d,p). Because enlargement of the is not responsible for the anomeric effect, we are not denying the basis set reduces the relative energies at the correlated MP2 level existence of this important interaction within molecules. This is quite more than at the uncorrelated HF level, we conclude that electron similar to the case of ethane, for which it was found that hypercon- correlation increases the preference for a gauche conformer. This jugation between two methyl groups makes a secondary contribution is due to the dispersion effect, which favours close contact (0.7–0.8 kcal mol21) to the rotation barrier (3.0 kcal mol21), but in between functional groups in gauche conformers. both eclipsed and staggered conformers there are still considerable As our interests lie in the n s* hyperconjugative interactions, hyperconjugative interactions (5.6–7.3 kcal mol21)26. we built the BLW of the primary Lewis structure where all electrons are strictly localized on their respective functional groups. In other Substituted tetrahydropyrans (C5OH9Y; Y 5 F, OH, Cl, NH2 and words, all electron delocalization effects are deactivated. For the CH3). Among the five substituted tetrahydropyrans, 2-fluorotetra- case of dimethoxymethane (for which the partition scheme is out- hydropyran, 2-chlorotetrahydropyran and 2-tetrahydropyranol lined in the Supplementary Information), we localized the C2H favour the a-anomers, thus exhibiting a significant anomeric effect, bonds within their own methyl or methylene groups, while constrain- as shown in Table 2. 2-Aminotetrahydropyran and 2-methyltetra- ing the oxygen lone electron pairs to individual oxygen atoms. Finally, hydropyran, however, are more stable with the b-anomer. In each C2O s bond was strictly localized between the two bonding particular, the reverse anomeric effect in 2-aminotetrahydropyran atoms. Nine blocks were defined, each containing an even number is well known both computationally and experimentally, and steric of electrons. Any MO was expanded within only one block or sub- repulsions have been cited as the cause20. This indicates that the group and doubly occupied. Energies of the strictly localized struc- dipole of the substituent as shown in Fig. 2 is not determined by Del tures, E(FL), and the regular HF energy E(C )arelistedin the polarity of the C12Y bond alone, but also by the lone pair(s) Table 1. The energy difference DE between BLW and HF represents and other atoms within the substituent group. Electron correlation the magnitude of electron delocalization. However, it should be noted is more pronounced in a-anomers than in their corresponding that DE includes not only the vicinal n s*ands s* hypercon- b-anomers, indicating the stabilizing role of dispersion in more jugative interactions, but also the geminal interactions among the crowded conformers. Overall, the strength of the anomeric effect is bonds sharing common apex atoms45. The hyperconjugative and dictated by the identity of the substituents in the following order, geminal interactions occur simultaneously and in general cannot F . Cl . OH . NH2 . CH3, in agreement with previous findings. effectively be separated, but the latter are essentially conserved in To evaluate the impact of hyperconjugation on the anomeric different conformers. Computational results with the basis sets 6- effect, from both the ring oxygen and the substituent, we computed 31G(d) and 6-311 þ G(d,p) are presented in Table 1. the delocalization energy by partitioning a substituted tetrahydro- Although a modest basis set effect was observed, the most signifi- pyran (C5OH9Y) into six overlapping blocks (see Supplementary cant finding is that the DEs are very comparable for all conformers. Information), one including the cyclic C5H9 fragment with 36 elec- In fact, the overall delocalization stabilization in the trans conformer trons, two corresponding to the two C2O s bonds, with two elec- is even slightly greater than in the gauche conformer. For example, trons for each block, the fourth consisting of the two lone pairs on

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Table 3 | Energy differences (DEab) between axial and equatorial conformations with decomposed contributions from electron delocalization (DEdel), steric effects (DEs) and dispersion effects (DEdisp).

21 21 21 21 Y DEab (kcal mol ) DEs (kcal mol ) DEdel (kcal mol ) DEdisp (kcal mol ) Substituted tetrahydropyran F 3.41 3.46 20.66 0.61 OH 1.32 2.19 21.42 0.55 Cl 2.69 1.76 0.70 0.33

NH2 22.73 22.45 20.61 0.34 CH3 23.30 22.31 21.18 0.18 Substituted cyclohexane F0.0920.07 20.01 0.17 OH 20.40 20.60 20.09 0.29 Cl 20.96 21.00 20.17 0.21

NH2 21.71 22.02 20.03 0.33 CH3 22.19 22.23 20.16 0.19 the oxygen, the fifth consisting of the s bond between the anomeric collectively cause the scenario shown by the EDD maps. To centre C1 and its linking atom from the substituent group Y, and the distinguish the n sCO* hyperconjugative interactions from the final sixth block comprising the substituent group alone. Clearly, the geminal interactions, we used the dimethyl ether as a reference BLW FL for the electron-localized diabatic state deactivates not only and similarly computed the DE with a wavefunction of five the hyperconjugative interactions from the oxygen lone pairs to the blocks. For the MP2/6-31 þ G(d) geometry, the DE in dimethyl substitute group Y, but also the geminal interactions from these two ether is 27.1 or 28.6 kcal mol21 with the basis sets 6-31 þ G(d) or groups as well as the exo-anomeric effect from Y to the ring. Table 2 6-311 þ G(d,p). Because the antibonding CO orbital is much listed the energies of the diabatic states for all five substituted tetra- lower in energy than the antibonding CH orbital, the fact that the hydropyrans. BLW computations revealed no change in the relative delocalization energy in dimethoxymethane is nearly twice that of stability between a- and b-anomers for all species following the DE in dimethyl ether suggests that both the n sCO* and n quenching of electron delocalization, once again proving that hyper- sCH* hyperconjugative interactions are fairly weak and the n conjugation has no role in influencing the anomeric effect. In fact, sCO* hyperconjugation thus cannot be the cause of the with the exception of 2-chlorotetrahydropyran, for which the exo- anomeric effect. anomeric effect from chlorine favours the a-anomer, the DEs in Similarly, for 2, both oxygen and fluorine delocalize their lone b-anomers are even slightly higher than in the corresponding a- pairs to the adjacent bonds (C2F and C2O), resulting in the anomers. These findings are consistent with the above studies on reduction of electron densities on oxygen and fluorine but the dimethoxymethane. Of the five substituted tetrahydropyrans, the enhancement of these bonds. The EDD maps in Fig. 3c for 3, delocalization energies in four species (Y ¼ F, OH, Cl, NH2)are however, are more illustrative, confirming the hyperconjugative 21 comparable within a narrow range of 35–37 kcal mol . The low interactions from oxygen lone pairs to the C12CH3 bond. As delocalization stabilization in 2-methyltetrahydropyran is obviously there is very little electron density change in the methyl group, in due to the lack of lone electrons in the substituted group. The capa- accordance with the weak hyperconjugative capability, there is an bility for electron delocalization from the substituted groups can be obvious increase in the electron density in the C12CH3 bond, justified from the computations for tetrahydropyran. BLW compu- and this electron flux must predominantly come from the oxygen, 21 tations at the same level resulted in a DE value of 22.7 kcal mol . indicating n sCC* hyperconjugative interactions. Taking the difference in DEs for 2-methyltetrahydropyran and tetra- The confirmation of hyperconjugative interactions, however, hydropyran, it is possible to estimate that hyperconjugation between does not indicate that they should be responsible for the anomeric the methyl group and the ring is about 7–8 kcal mol21. This is very effect, as our BLW computations have profoundly demonstrated. close to the hyperconjugative stabilization energy in ethane26. Further evidence can be taken from a comparison with substituted cyclohexanes (C6H11Y), as the energy difference between the equa- Discussion torial and axial conformations of substituted cyclohexane is often The BLW computations for dimethoxymethane and substituted tetra- taken as a reference with which to measure the magnitude of the hydropyrans confirm the persistence of the anomeric effect, even anomeric effect18. However, the greater length of the C2C bond when all intramolecular electron delocalization is quenched. Thus, in C6H11Y compared with the C2O bond in C5OH9Y relieves the the hyperconjugation model is unable to explain the anomeric effect. steric repulsion and consequently underestimates the anomeric To visualize the electron delocalization effect within the molecu- effect46,47. (Computational results for substituted cyclohexanes are lar system, we plotted the electron density difference (EDD) between presented in the Supplementary Information.) Owing to steric the delocalized and localized wavefunctions. Figure 3 shows the effects, except for F, which is neutral with regard to axial or equator- EDD maps for the three conformers of dimethoxymethane (1) ial positions, all substituents prefer the equatorial position. It is and the a- and b-anomers of 2-fluorotetrahydropyran (2) and noteworthy that the electron delocalization effect in the a- and 2-methyltetrahydropyran (3), where the red colour denotes a gain b-anomers of each molecule is particularly close. and the blue refers to a loss of electron density. For 1, the delocali- To explain the origin of the anomeric effect in substituted tetra- zation reduces the electron density around the oxygen atoms, but hydropyrans, we decomposed the axial–equatorial energy differences the central methylene group also loses electrons. All these lost elec- (DEab) at the MP2 level into three components, namely electron trons flow to the C2O bond regions. If there were only n sCO* delocalization (DEdel), steric effect (DEs) and dispersion (DEdisp), as interactions, we would observe a significant electron density increase in the two central C2O bonds. However, the EDD maps = ( )− ( )= ( )− ( )+ DEab Eb MP2 Ea MP2 Eb HF Ea HF DEdisp in Fig. 3a show the comparable changes in central and terminal = ( )− ( )+ + C2O bonds. As a consequence, both the geminal delocalization, Eb BLW Ea BLW DEdel DEdisp which can be described in terms of re-hybridization of the oxygen = D + D + D ( ) orbitals, and hyperconjugative delocalization are involved and Es Edel Edisp 1

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A The steric energy term consists of both the electrostatic interaction where V is a successive product of nA occupied spin–orbitals in subgroup A: (stabilizing or destabilizing) and Pauli repulsion. Table 3 contains VA = cAacAb ···cA b (4) the energy terms for substituted tetrahydropyrans and cyclohexanes. 1 1 nA /2 It is clear that the steric effect dominates in the conformational preferences for all cases. Except for 2-chlorotetrahydropyran, for which If we allow all orbitals to expand in the whole space of primitive orbitals, equation (3) will be equivalent to the familiar Hartree–Fock (or Kohn–Sham with density a significant exo-anomeric effect favours the a-anomer, all species functional theory) wavefunction CDel, corresponding to a delocalized, adiabatic exhibit stronger hyperconjugative interactions in the equatorial con- state, which is implicitly a superposition of all electron-localized states. If we choose formers than in the axial conformers. In addition, the dispersion the resonance structure L as the most stable resonance structure, the energy Del effect stabilizes all axial forms, although the magnitude is small. difference between FL and C is generally defined as the delocalization energy One further support for the electrostatic explanation comes from (or resonance energy, RE): analysis of the relative orientations of the local dipoles in substituted DE = E(F )−E(CDel)(5) tetrahydropyrans (see Fig. 2). We used model molecules L CH32O2CH3 and CH32Y at the exact positions of the substituted In applications, however, we can generally define blocks in a much more flexible way tetrahydropyran to compute the orientations of the local by focusing on the specific delocalization effect. For instance, a negative dipoles (listed in the Supplementary Information). The data are in hyperconjugation can be studied by localizing the lone pairs and adjacent bonds agreement with the conformational preferences. only in FL; consequently the DE measures the hyperconjugative interactions In summary, to evaluate the stabilization energy due to electron between these lone pairs and surrounding bonds. In this work, all geometries were optimized at the MP2/6-31 þ G(d) level and delocalization (or electron transfer) at the quantitative level, it is subsequent BLW calculations were carried out using the Xiamen Valence Bond essential to derive the wavefunction for the electron-localized diabatic (XMVB) program48 and a modified version of GAMESS49. state for reference. We have proposed a general BLW method, which can be regarded as the simplest variant of ab initio VB theory. Received 28 January 2010; accepted 17 May 2010; Applications of the BLW method to dimethoxymethane and substi- published online 4 July 2010 tuted tetrahydropyrans demonstrated that electron delocalization slightly favours the trans conformer of dimethoxymethane and equa- References torial anomers of substituted tetrahydropyrans (C OH Y, Y ¼ F, OH, 1. Edward, J. T. Stability of glycosides to acid hydrolysis. Chem. Ind. (Lond.) 5 9 1102–1104 (1955). NH2 and CH3) with the exception of 2-chlorotetrahydropyran. 2. Lemieux, R. U. & Chu¨,P.in133rd National Meeting of the American Chemical However, even for the latter, the stabilization energy due to the Society 31N (American Chemical Society, 1958). hyperconjugation from chlorine to the ring accounts for only a 3. Eliel, E. L. Conformational analysis in heterocyclic systems: recent results and small portion of the axial–equatorial energy difference. Thus, our applications. Angew. Chem. Int. Ed. Engl. 11, 739–750 (1972). 4. Szarek, W. A. & Horton, D. (eds) Anomeric Effect: Origin and Consequences BLW computations do not support the currently popular hypercon- (American Chemical Society, 1979). jugation explanation for the anomeric effect. Energy analyses 5. Kirby, A. J. Anomeric Effect and Related Stereoelectronic Effects at Oxygen revealed that the steric effect, which includes both electrostatic inter- (Springer-Verlag, 1983). action and Pauli repulsion, dominates conformational preferences. 6. Deslongchamps, P. Stereoelectronic Effects in Organic Chemistry (Elsevier, 1983). An alternative explanation such as the electrostatic model must 7. Tvaroska, I. & Bleha, T. Anomeric and exoanomeric effects in carbohydrate chemistry. Adv. Carbohydr. Chem. Biochem. 47, 45–123 (1989). therefore be introduced to interpret the anomeric effect. 8. Thatcher, G. R. (ed.) The Anomeric Effect and Associated Stereoelectronic Effects (American Chemical Society, 1993). Methods 9. Juaristi, E. & Cuevas, G. The Anomeric Effect (CRC Press, 1995). The significant difference between the MO and VB theories is that, by definition, the 10. Smith, M. B. & March, J. March’s Advanced Organic Chemistry (Wiley, 2007). latter can construct wavefunctions for individual resonance structures or diabatic 11. Anderson, C. B. & Sepp, D. T. Conformation and the anomeric effect in states where each pair of electrons is strictly localized on one atom or two bonding 2-halotetrahydropyrans. J. Org. Chem. 32, 607–611 (1967). atoms, with the expense of increased computational costs due to the non- 12. Ha, S. H., Gao, J., Tidor, B., Brady, J. W. & Karplus, M. Solvent effect on the orthogonality of the orbitals. In contrast, MO theory allows all orbitals to delocalize anomeric equilibrium in D-glucose—a free-energy simulation analysis. J. Am. over the whole system with the constraint of orthogonality, and is therefore unable to Chem. Soc. 113, 1553–1557 (1991). define diabatic states. In VB theory, the wavefunction for a resonance structure L 13. Cramer, C. J. Anomeric and reverse anomeric effects in the gas phase and (here we assume closed-shell cases, that is, N is an even number) can be expressed by aqueous solution. J. Org. Chem. 57, 7034–7043 (1992). a Heitler–London–Slater–Pauling (HLSP) function as 14. Fuchs, B., Schleifer, L. & Tartakovsky, E. Structure and conformation of heterocycles .14. probing the anomeric effect—the structural criterion. New J. = ˆ ··· ( ) ( )− ( ) ( ) ( ) Chem. 8, 275–278 (1984). FL NLA w1w2w3 wN a i b j b i a j 2 ij 15. Perrin, C. L., Armstrong, K. B. & Fabian, M. A. The origin of the anomeric effect: conformational analysis of 2-methoxy-1,3-dimethylhexahydropyrimidine. ˆ where NL is the normalization constant, A is an antisymmetrizer and in the above J. Am. Chem. Soc. 116, 715–722 (1994). resonance structure L, two electrons on orbitals wi and wj form a chemical bond. The 16. Box, V. G. S. The anomeric effect of monosaccharides and their derivatives. overall wavefunction C is a superposition of all independent resonance structures, Insights from the new QVBMM molecular mechanics force field. Heterocycles and the delocalization energy can be generally ‘obtained by subtracting the actual 48, 2389–2417 (1998). energy of the molecule in question from that of the most stable contributing 17. Fuchs, B., Ellencweig, A., Tartakovsky, E. & Aped, P. Structure and conformation structure’29. Apparently, each HLSP can be expanded into 2N/2 Slater determinants, of heterocycles. 15. Solvent polarity and the anomeric effect. Angew. Chem. Int. so the electron correlation is taken into account. Ed. 25, 287–289 (1986). One way to dramatically simplify HLSP functions yet retain the VB 18. Juaristi, E. & Cuevas, G. Recent studies of the anomeric effect. Tetrahedron 48, characteristics of localization is to make two bonding orbitals wi and wj equal but 5019–5087 (1992). strictly localized on their bonding area. As such, equation (2) will be reduced to only 19. Vila, A. & Mosquera, R. A. Atoms in molecules interpretation of the anomeric one Slater determinant. A general extension of this idea is the BLW method33,34,in effect in the OCO unit. J. Comput. Chem. 28, 1516–1530 (2007). which all primitive orbitals and electrons are partitioned into K blocks (or groups), 20. Salzner, U. & Schleyer, P. v. R. Ab initio examination of anomeric effects in and each MO is allowed to expand only in one block. Although block-localized tetrahydropyrans, 1,3-dioxanes and glucose. J. Org. Chem. 59, 2138–2155 (1994). MOs (BL-MOs) in the same block are constrained to be orthogonal, BL-MOs 21. Reed, A. E., Curtiss, L. A. & Weinhold, F. Intermolecular interactions from a between different blocks are non-orthogonal. However, the above partition is not natural bond orbital, donor2acceptor viewpoint. Chem. Rev. 88, 899–926 (1988). enough for general cases in which certain primitive orbitals may be involved in 22. Pophristic, V. & Goodman, L. Hyperconjugation not steric repulsion leads to the more than one block. A further generalized BLW method defines each block with staggered structure of ethane. Nature 411, 565–568 (2001). a subspace of all primitive orbitals, without imposing the limit that any primitive 23. Bickelhaupt, F. M. & Baerends, E. J. The case for steric repulsion causing the orbital can appear only once in blocks. In such a way, equation (2) can be simplified staggered conformation of ethane. Angew. Chem. Int. Ed. 42, 4183–4188 (2003). to the form 24. Weinhold, F. Rebuttal to the Bickelhaupt–Baerends case for steric repulsion causing the staggered conformation of ethane. Angew. Chem. Int. Ed. 42, = ˆ { 1 2 ··· K }() FL NLA V V V 3 4188–4194 (2003).

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